You could call it 'cherry picking' - the selection of terms/facts or data which suit your purpose, but might not reflect the whole picture. It could also be called 'taking it out of context' - using a phrase or figure in a manner other than which it was written/intended.

As has been mentioned, interpretation bias and/or confirmation bias work as well.

Confirmation bias/cherry picking is about choosing the bits of your data that support the conclusion that you want to find. If this is the case, then your conclusion is still backed by evidence, but the illegitimate step was in the selective use of data (ignoring things that don't support your conclusion), not in the interpretation.

The other terms like reading into', Hineininterpretation, eisegesis etc. are about making claims that aren't supported by the data, they've been shipped in by you. The data might be complete and adequate, but it's the conclusions you draw from it that are illegitimate - in which case, it's not evidence for your claim.

Since all interpretation is a kind of 'reading into' (since data doesn't interpret itself) it seems to me that there are two more possibilities here: importing data that isn't there to support a conclusion, or failing to consider other interpretations

For example, you see a colleague that you haven't seen for a while, and ask where she's been. She replies, "I've been on leave". If you then say "Oh, how lovely, did you have a nice time?" then you've interpreted "leave" as meaning "holiday", even though there are other kinds of leave - compassionate leave and sick leave, for example, where "did you have a nice time?" is an inappropriate response.

There are two possible errors

a) You added the concept "holiday" to the term "leave" even though it wasn't in the data, then drew your conclusion

b) You interpreted "leave" as holiday, without considering first whether there were other equally valid interpretations to choose from.

In which case, the phrase would be jumping to conclusions

A favourite term of mine, though I'm not sure who coined it is armchair hermeneutics - though that's not really the same thing, that's just reasoning from your chair without bothering to support your claims with evidence. There is an element of that here, though.

"Motivated Reasoning" is a term that I've seen recently to describe the process of interpreting information in order to support a predetermined conclusion. Various methods can be used in the process, including confirmation bias and cherry-picking as were mentioned in other answers, but there are numerous other techniques as well. Motivated Reasoning seems to be a descriptive name for the overall process, regardless of how it is done.